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# Parameter Estimation/Tweaking Problems solved with Calculus Programming _________

This section shows how to solve equations of the following form:.

Uxx = f(x, U, Ux; a, b, c, ...)
where the parameters (a, b, c, etc.) are tweaked/estimated

This is a general form of a partial differential equation. To tweak parameters a, b, c, etc. through Calculus-level programming we replace the following code for the "call xAxis" statement in these examples:

 ``` a = 1: b = 1: c = 1 ! Initial Estimations find a, b, c, ... In xAxis ooo to optimize xyz where 'optimize' may be the term 'maximize', 'minimize', 'match', or 'extreme'; 'match' implies your objective function converges to zero.``` plus, add an objective function in your math model. ooo xyz = some function of parameters a, b, c, etc.

This 'find' statement will vary your parameters until 'xyz' is optimal. Thus meeting your objective requirements.

Enjoy learning Calculus-level Programming!

### Example Parameter Estimation/Tweaking Problem Source Code:

#### A Parameter Estimation/Tweaking Problem:

``````      global all
problem nonLinPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: non-linear PDE (1D) Parameter
C --- Estimation Problem solved.
C ------------------------------------------------------------------------
C
C User parameters ...
!       rho = ...
e0 = 8.854187817e-12 ! F/m or A2 s4 kg-1m−3 permittivity of free space
!        ipoints = 10          ! grid pts. over x-axis
C
C x-parameter initial settings: x ==> i
!        xFinal =  1:    xPrint = xFinal/ipoints:       ip = ipoints
pi= 4*atan(1)
C
a = 1:	b = 1:    c = 1		! Initial Estimations
find a, b, c, ... ;   in xAxis;    by ?solver?;   to ?optimize? xyz
end
model xAxis
C ... Integrate over x-axis
C
x= 0:    xPrt = xPrint:      dx = xPrt / 10
Initiate janus;  for PDE;
~       equations Uxx/Ux, Ux/U;  of x;  step dx;  to xPrt
!        U = 1.23		! @ x = 0 ... an Initial Value Problem
do while (x .lt. xFinal)
Integrate PDE;  by janus
if( x .ge. xPrt) then
print 79, x, U, Ux, Uxx
xPrt = xPrt + xPrint
end if
end do
79     format( 1h , f8.4, 2x, 10(g14.5, 2x))
end
model PDE                         ! Partial Differential Equation
Uxx = -rho/e0 * (1.23 + sin( Ux * pi) - .543) ooo	! your Diff Eq goes here
xyz = function of parameters to be optimized ...
! e.g.   xyz = U + c - b / (1 + Uxx * c**2)
end
``````

### Example Parameter Estimation/Tweaking Problem Output:

``selected output goes here ...                ``